Question

What is the amplitude, period, and phase shift of f(x) = -4 sin(2x + π) - 5?

Answer 1

\[f(x)=a \cdot \sin(b(x-c))+d \\ |a| \text{ is amp } \\ \frac{2 \pi}{|b|} \text{ is period } \\ c \text{ is phase shift } \\ d \text{ is vertical shift }\]

lets have example showing these things:


So we know
\[g(x)=\sin(x) \text{ on } [0,2 \pi]\]
looks like:


If we have
\[h(x)=2 \cdot g(x) \text{ then all of are } y \text{-values get times by 2 on our } g \text{-graph} \]
notice the change here:

so basically the amp number tells us how far we are away from the center axis (center axis can change if there is a vertical shift)

say we have:

\[h(x)=2 \cdot g(x)+3 \]
Well we still had all the y's on the g-graph get multiply by 2 and not we are going to add 3 to each those results
so the center axis is now y=3 instead of what is was in the previous example which was y=0 or aka the x-axis




now we can also mess with the period by missing with the inside of the sin( ) function

that is if we have
\[v(x)=\sin(\pi x)\]
so the period for g(x)=sin(x) was 2pi

so to find the period of v we need to look at the inside of the sine function and see what is next to x and divide 2pi by it

so we see the period is 2
which actually makes sense if you think about it